Question: Solve for $x$ and $y$ by deriving an expression for $x$ from the second equation, and substituting it back into the first equation. $\begin{align*}2x+7y &= 9 \\ 9x+4y &= -9\end{align*}$
Answer: Begin by moving the $y$ -term in the second equation to the right side of the equation. $9x = -4y-9$ Divide both sides by $9$ to isolate $x$ $x = {-\dfrac{4}{9}y - 1}$ Substitute this expression for $x$ in the first equation. $2({-\dfrac{4}{9}y - 1}) + 7y = 9$ $-\dfrac{8}{9}y - 2 + 7y = 9$ Simplify by combining terms, then solve for $y$ $\dfrac{55}{9}y - 2 = 9$ $\dfrac{55}{9}y = 11$ $y = \dfrac{9}{5}$ Substitute $\dfrac{9}{5}$ for $y$ in the top equation. $2x+7( \dfrac{9}{5}) = 9$ $2x+\dfrac{63}{5} = 9$ $2x = -\dfrac{18}{5}$ $x = -\dfrac{9}{5}$ The solution is $\enspace x = -\dfrac{9}{5}, \enspace y = \dfrac{9}{5}$.